A finite element model for protein transport in vivo
- Kouroush Sadegh Zadeh^{1, 2}Email author,
- Howard C Elman^{1},
- Hubert J Montas^{2} and
- Adel Shirmohammadi^{2}
https://doi.org/10.1186/1475-925X-6-24
© Sadegh Zadeh et al; licensee BioMed Central Ltd. 2007
Received: 19 April 2007
Accepted: 28 June 2007
Published: 28 June 2007
Abstract
Background
Biological mass transport processes determine the behavior and function of cells, regulate interactions between synthetic agents and recipient targets, and are key elements in the design and use of biosensors. Accurately predicting the outcomes of such processes is crucial to both enhancing our understanding of how these systems function, enabling the design of effective strategies to control their function, and verifying that engineered solutions perform according to plan.
Methods
A Galerkin-based finite element model was developed and implemented to solve a system of two coupled partial differential equations governing biomolecule transport and reaction in live cells. The simulator was coupled, in the framework of an inverse modeling strategy, with an optimization algorithm and an experimental time series, obtained by the Fluorescence Recovery after Photobleaching (FRAP) technique, to estimate biomolecule mass transport and reaction rate parameters. In the inverse algorithm, an adaptive method was implemented to calculate sensitivity matrix. A multi-criteria termination rule was developed to stop the inverse code at the solution. The applicability of the model was illustrated by simulating the mobility and binding of GFP-tagged glucocorticoid receptor in the nucleoplasm of mouse adenocarcinoma.
Results
Conclusion
The developed strategy is an efficient approach to extract as much physiochemical information from the FRAP protocol as possible. Well-posedness analysis of the inverse problem, however, indicates that the FRAP protocol provides insufficient information for unique simultaneous estimation of diffusion coefficient and binding rate parameters. Care should be exercised in drawing inferences, from FRAP data, regarding concentrations of free and bound proteins, average binding and diffusion times, and protein mobility unless they are confirmed by long-range Markov Chain-Monte Carlo (MCMC) methods and experimental observations.
Background
Transport of mass, energy, and momentum has a crucial role in many branches of science and engineering. In biological systems, transport phenomena are central to the biological processes that take place in different parts of organisms. They determine the behavior and function of cells, tissues, and organs, and regulate interactions between synthetic agents (e.g. drugs) and recipient targets. These phenomena are crucial elements in the design and use of biosensors, high density cell culture, filtration units for kidney dialysis, heart-lung bypass machines, and membrane oxygenators in human medical care and ion selective electrodes, pH-meters, electrical conductivity meters, and time domain reflectometery used in biosystem analysis. Transport processes are critical in the removal of toxins from the blood, remediation of impaired water bodies (sources of waterborne diseases), and bioremediation of contaminated landscape [1, 2].
Fluorescence Recovery after Photobleaching (FRAP) is one of the most widely used experimental protocols to study biological transport processes such as diffusion and reaction [3–30]. FRAP is a straightforward technique used to monitor the movement of fluorescence molecules. These molecules can absorb light of one wavelength (blue for instance) and emit light of another (e.g. green). However, if exposed to repeated cycles of excitation-emission, they lose their ability to emit fluorescence. This phenomenon is called "photobleaching" or "photochemical bleaching" [24]. In this technique a small region of living cell containing Green Fluorescent Protein (GFP)-tagged protein is exposed to a brief but intense laser beam, produced by a laser scanning confocal microscope, to irreversibly inactivate fluorescence emission in that region. Before exposure to the light, the living cell is in equilibrium with a uniform population of fluorescence [25]. Photobleaching creates two different populations of fluorescence molecules, which are spatially separated at the beginning of the experiment. Unbleached molecules from the undisturbed area move toward the bleached region and the rate of fluorescence recovery is measured as a function of time. The result is a noisy graph known as a recovery curve. However, because of the noise, the original graph by itself is not suitable for quantitative study of the dynamics of living cells. The FRAP community generally uses a data processed normalized average fluorescence recovery curve that has less noise. By analyzing the recovery curve, one can quantify how many fluorescent photons return to the bleached area in comparison to the amount of light that was there before photobleaching. This is known as percent recovery. The other question that can be addressed is that of how fast the fluorophores move toward the bleached area. This is a measure of the free molecular diffusion coefficient of the bio-macromolecule under study.
One of the first attempts to estimate bio-macromolecule mass transport and binding rate parameters using in vivo information was carried out by Kaufmann and Jain [13]. Sprague et al. [25] developed a diffusion-reaction model to simulate FRAP experiment but the solution is in Laplace space and requires numerical inversion to return to real time. The model presented recently by Lele et al. [26] properly respects cell boundaries but is in the form of a Fourier-Bessel series and can suffer from Gibbs phenomena. Carrero et al. (2003, 2004) presented an excellent review on the effects of boundary conditions, influence of the membrane, and the location of the photobleaching on the estimation of diffusion coefficients for diffusing biomolecules in a bounded domain [24, 28]. They showed that overestimations or underestimations can result from ignoring this influence [24]. Beaudouin et al. [29] used a diffusion-reaction models to study mobility of five chromatin-interacting proteins inside living cells. They found that transient interactions are common for chromatin proteins. Individual proteins locally sample chromatin for binding sites rather than diffusing globally followed by binding at random nuclear positions. They concluded that complementary procedures are needed to measure transient biochemical interactions in living cells.
Although experimental methods are representative of the biological system, they are expensive, tedious and time-consuming. An alternative approach is to use mathematical modeling. In this regard, several sophisticated mathematical models have been developed to predict and simulate the fate and transport of drugs and bio-macro-molecules in biological systems. However, the use of these models is not an easy task since they contain numerous parameters that need to be determined before the model(s) can be used for the considered situation. The success of model predictions depends largely on the proper representation of relevant processes, uncertainty in model parameters [31, 32], and parameter identification which is a critical step in the modeling process. Difficulties in model calibration and parameter identification are quite common in modeling mass transport problems in biological systems [1, 30].
The main objective of this paper is to develop a mass-lumped Galerkin-based finite element model (FEM) to solve a system of two partial differential equations governing protein transport and binding in living cells and couple it with the Osborne-Moré [33, 34] extended version of the Levenberg-Marquardt [35, 36] algorithm and an experimental data set, obtained by the Fluorescence Recovery after Photobleaching (FRAP) technique, to quantify bio-macromolecule diffusion coefficient and binding rate parameters. The applicability of the developed FEM-based inverse modeling strategy is illustrated by simulating the mobility and binding of GFP-tagged glucocorticoid receptor in the nucleoplasm of mouse adenocarcinoma.
Methods
Direct problem
Since a circular bleach spot was used to bombard the cell and to track the fluorescently tagged biomolecules inside bleach spot during the time course of a FRAP experiment, system (1) was written in cylindrical coordinate system.
The initial condition implies that photochemical bleaching inactivates the fluorescence tag on the biomolecules in the bleached area but does not change the concentrations of free and bound biomolecules or vacant binding sites. The boundary conditions imply that the diffusive biomolecule flux is zero at the center of the bleach spot and at the cell or nucleus membrane during time course of a FRAP experiment [30].
in which N is total number of nodes in the spatial domain, ϕ _{ j }(ρ) are the selected linear basis functions, and F _{ j }(t) and C _{ j }(t) are the associated time-dependent unknown coefficients that represent the solution of Equation (1) at nodes within the domain.
in which Ω_{ e }is the domain of element and q _{ fn }and q _{ cn }are fluxes of the free and bound bio-macromolecules across the boundary out of the element, respectively.
in which ρ _{ e }is the radial position of the centroid of element e.
Parameter optimization
The maximum of the likelihood function must satisfy the set of equations ∂ ln L(p)/∂p = 0. When the error covariance matrix is known, maximization of equation (8) is equivalent to the minimization of the following weighted least square problem [1]: ϕ(p) = [(U* - U(p))^{ T } V^{-1}(U* - U(p))]
in which p* is parameter vector containing a priori information, $\widehat{p}$ is the corresponding predicted parameter vector, and V _{ p }is the covariance matrix for the parameter vector. This kind of optimization is known as Bayesian estimation. The second term in equation (10), which is sometimes called the plausibility criterion [1] insures that the optimized values of the parameters remain in some feasible region around p*. Matrices V and V _{ p }, which are sometimes called weighting matrices, provide information about the measurement accuracy as well as any possible correlation between measurement errors and between parameters.
where r is the residual (differences between the observed and predicted state variable) column vector.
Minimization of equation (11) was carried out iteratively by first starting with an initial guess of parameter vector, {p^{(k)}} and updating it at each iteration until the termination criteria were met [1, 31, 32]: p^{(k+1)}= p^{(k)}+ α^{(k)}Δp^{(k)} (12)
where α^{(k)}is a scalar step length and Δp^{(k)}is the direction of the search or step direction [41].
where λ is a positive scalar known as Marquardt's parameter or Lagrange multiplier [36], J is the Jacobian or sensitivity matrix, and D is a positive definite scaling matrix that ensures the descent property of the algorithm even if the initial guess is not "smart". For non-zero values of λ, the Hessian approximation is always a positive definite matrix, which ensures the descent property of the algorithm [41].
A combination of "one-sided" and "two-sided" finite difference methods [30–32] was used to calculate the partial derivatives of the state variable with respect to model parameters and to construct the Jacobian matrix. The "One-sided" finite difference method estimates the partial derivatives of the state variable with respect to model parameters by solving the forward problem (Eq. 1) p+1 times (p is number of parameters to be estimated). On the other hand, the "two-sided" finite difference method estimates the partial derivatives of the state variable with respect to model parameters by solving the forward problem (Eq. 1) 2p+1 times. At the early stages of the optimization, where the search is far from the solution, the "one-sided" finite difference scheme, which is computationally cheap but not as accurate as the "two-sided" approach, was used. As the optimization proceeds in the descent direction, the algorithm switches to a more accurate but computationally expensive approach in which the partial derivatives of the state variable with respect to the model parameters are calculated using a two-sided finite difference scheme. The switch was made when ϕ(p) ≤ 1 × 10^{-2}. A detailed description of the procedure to update the Jacobian matrix is presented in [1, 30].
In order to update λ at each iteration, the optimization starts with an initial parameter vector and a large λ(λ = 1). As long as the objective function decreases in each iteration, the value of λ is reduced. Otherwise, it is increased. The approach avoids calculation of λ and step length in each iteration and is therefore computationally cheap. A detailed description of the code for updating λ is given in [31].
Stop
else
Continue Optimization Loop
end
where ${\nabla \phi (p)|}_{p=\widehat{p}}$ is the gradient of the penalty function at solution, $\frac{\Delta \phi (p)}{\phi (p)}$ is relative changes in the magnitude of the parameters in two consecutive iterations, and δ _{1}, δ _{2}, and δ _{3} are user defined small values.
where I _{ i }and ${\widehat{I}}_{i}$ are the observed and predicted total normalized average fluorescence intensity, F + C, inside bleached area during time course of FRAP experiment, respectively and N is the number of observations on FRAP time series.
Experimental Studies
A FRAP experiment was conducted on the mouse adenocarcinoma cell line 3617 at the Laboratory of Receptor Biology and Gene Expression, National Cancer Institute-National Institutes of Health, Bethesda, MD (McNally, personal communication). This data set consists of 43 fluorescent recovery values gathered in the course of a 20-second FRAP experiment and post processed to reduce noise. The developed inverse modeling strategy was then used to quantify mass transport and binding rate parameters of GFP-tagged glucocorticoid receptor.
Results and Discussion
Model validation
While Figure 1a shows spatial and temporal distributions of free GFP-GR, F(ρ, D _{ f }, ${K}_{a}^{\ast}$, K _{ d }, t), inside and outside bleach spot after photochemical bleaching during time course of a FRAP experiment, Figure 1b presents distributions of bound protein, C(ρ, D _{ f }, ${K}_{a}^{\ast}$, K _{ d }, t), at times of 0, 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1, and 2 seconds. Both Figures indicates that there is excellent agreement between the analytic solution and numerical simulator. Spatial and temporal distributions of total fluorescently tagged GFP-GR, [F + C](ρ, D _{ f }, ${K}_{a}^{\ast}$, K _{ d }, t), inside and outside the bleached area are given in Figure 1c, which presents excellent agreement between the analytic and numerical solutions. The same time range was used to perform the comparison. As Figure 1d indicates, there is excellent agreement between the two solutions in simulating the average normalized fluorescence intensity within the bleach spot during time course of the FRAP experiment.
Model calibration
The results of parameter estimation for GFP-GR using an experimental time series obtained by the Fluorescence Recovery after Photobleaching (FRAP) technique.
run | D _{ f }(μ m^{2} s^{-1}) | ${K}_{a}^{\ast}$ (s^{-1}) | K^{ d }(s^{-1}) | F _{ eq } | C _{ eq } | t _{ b }(ms) | t _{ d }(ms) | RMSE | R ^{2} |
---|---|---|---|---|---|---|---|---|---|
1* | 9.20 | 500 | 86.4 | 0.1473 | 0.8527 | 11.60 | 2 | 0.0255 | 0.9886 |
2 | 1.9049 | 1.0549 | 3.7657 | 0.7812 | 0.2188 | 265 | 0.948 | 0.0259 | 0.9897 |
3 | 1.2319 | 0.1113 | 12.0951 | 0.9909 | 0.0091 | 83 | 8985 | 0.0245 | 0.9903 |
4 | 3.4980 | 10.2501 | 8.3917 | 0.4502 | 0.5498 | 119 | 97.6 | 0.0275 | 0.9882 |
5 | 22.8472 | 369.7719 | 24.1606 | 0.0613 | 0.9387 | 41 | 2.7 | 0.0251 | 0.9898 |
6 | 81.9332 | 4785 | 75.4057 | 0.0155 | 0.9845 | 13 | 0.2 | 0.0233 | 0.9912 |
7 | 1.2160 | 0.7076 | 5057 | 0.9999 | 0.0001 | 0.2 | 1413 | 0.0245 | 0.9903 |
8 | 1.8034 | 0.5172 | 2.4778 | 0.8273 | 0.1727 | 403 | 1933 | 0.0259 | 0.9896 |
9 | 4.6451 | 35.3378 | 15.8084 | 0.3091 | 0.6909 | 63 | 28.3 | 0.0257 | 0.9895 |
10 | 4.4471 | 1.4531 | 2.0004 | 0.5792 | 0.4208 | 500 | 688 | 0.0321 | 0.9855 |
11 | 1.2014 | 0.00003 | 20.3473 | 1.0000 | 0.0000 | 49 | 3.3 × 10^{7} | 0.0246 | 0.9901 |
12 | 20.3662 | 928 | 62 | 0.0626 | 0.9374 | 16 | 1078 | 0.0233 | 0.9912 |
13 | 7.4662 | 101 | 23.3243 | 0.1876 | 0.8124 | 43 | 9901 | 0.0247 | 0.9904 |
14 | 1.2210 | 7.5124 | 1423 | 0.9947 | 0.0053 | 0.7 | 1333113 | 0.0245 | 0.9903 |
15 | 19.5330 | 160 | 13.9388 | 0.0801 | 0.9199 | 72 | 6250 | 0.0290 | 0.9865 |
16 | 1.2350 | 32.2818 | 1590 | 0.9801 | 0.0199 | 0.6 | 30977 | 0.0245 | 0.9902 |
17 | 11.0675 | 300.4231 | 40.4005 | 0.1185 | 0.8815 | 25 | 3329 | 0.0235 | 0.9910 |
18 | 9.1683 | 1524 | 237.1330 | 0.1346 | 0.8654 | 4 | 656 | 0.0239 | 0.9908 |
19 | 8.3273 | 145.5289 | 28.1600 | 0.1621 | 0.8379 | 35 | 6871 | 0.0241 | 0.9906 |
20 | 1.2275 | 23.0060 | 3478 | 0.9934 | 0.0066 | 0.3 | 43467 | 00245 | 0.9908 |
21 | 4.9839 | 653.6816 | 214.1402 | 0.2468 | 0.7532 | 5 | 1530 | 0.0239 | 0.9908 |
22 | 94.2711 | 1857.1133 | 27.6413 | 0.0147 | 0.9853 | 36 | 538.5 | 00247 | 0.9901 |
23 | 1.2584 | 21.1318 | 590 | 0.9654 | 0.0346 | 2 | 47322 | 0.0245 | 0.9903 |
24 | 4.5057 | 63.7910 | 26.8675 | 0.2964 | 0.7036 | 37 | 15676 | 0.0239 | 0.9908 |
Analysis of Table 1 indicates that the average binding time per vacant site, calculated by t _{ b }= 1/K _{ d }[25], ranges between 0.3 and 500 mili-seconds. Similarly, the average time for diffusion of GFP-GR from one binding site to the next, obtained by t _{ d }= 1/${K}_{a}^{\ast}$[46], varies over several orders of magnitude. The broad range of average binding time and average diffusion time for GFP-GR indicates that it is doubtful that one can infer an average time for macro-molecule diffusion and binding inside living cells from FRAP results alone.
This analysis explains the conflicting parameter values that may have been reported in literature for biomolecules using the FRAP protocol. Based on our findings, using experimental FRAP time series and coupling it with curve-fitting procedures can lead to misleading conclusions regarding binding reaction, slow or rapid mobility of biomolecules, and concentrations of free macromolecule, vacant binding sites and bound complex inside living cells and tissues.
In this study, the choice of a numerical approach rather than an analytic solution and a finite element approach rather than the finite difference scheme was made so that the parameter estimation could be readily extended to arbitrary initial and boundary conditions, complex domain geometry, and especially so that it could be extended to a ternary system of coupled nonlinear partial differential equations governing transport of free bio-macromolecule, bound complex, and vacant binding site where all three transport entities are moving species and it is impossible to obtain an exact solution for the systems of equations.
So far most of the FRAP studies have assumed an infinite domain to specify boundary conditions and to solve the system of partial differential equations governing biomolecule transport in-vivo [[13, 25, 29, 30] among many others]. This assumption is unrealistic in the context of living cells. In this study, we address and improve this shortcoming by specifying a finite domain and by formulating Neumann boundary conditions on the cell membrane.
Residual analysis
The inverse methodology used in this study is based upon the following assumptions: 1) residuals have a mean of zero, 2) residuals have constant variance, 3) residuals are uncorrelated, and 4) residuals are normally distributed. When these assumptions are met, the parameter optimization estimates poses optimal statistical properties [40–42]. When these conditions are not met the parameter optimization method may no longer produce optimal parameter estimates. Residuals, or errors in parameter optimization, are defined as the difference between the observed and simulated state variable(s). An analysis of the residuals is a useful and key technique to study possible trends, oscillations, and correlation of errors. It is also important in validating the assumptions on which the inverse modeling strategy rests.
To analyze the residuals, they were plotted against average normalized fluorescence intensity, $\overline{I}$(t), within the bleach spot during the time course of the FRAP experiment. Since the residuals are time and/or space series, their possible correlation was thoroughly analyzed. Different statistical measures such as error frequency analysis, normal probability plot, and hypotheses tests were explored to make decision about the residuals. The Student's t-test was used to test if the residuals have a mean of zero. Bartlett's test [47] was applied to determine if the residuals have constant variance. To test the normality of the residuals the Chi-square and Kolmogorov-Smirnov one sample tests were employed. Finally, the t-statistic [48] was used to test if the residuals are correlated.
in which $\overline{x}$, s, and N are the mean, standard deviation, and size of the sample (errors), respectively. μ _{0} is the mean of the population which is zero.
For -t _{ α/2}<t <t _{ α/2}the null hypothesis (mean is zero) cannot be rejected at the significance level of α. The rejection regions t < -t _{ α/2}or t _{ α/2}<t indicate that the null hypothesis can be rejected at the level of significance α.
The results of hypothesis test on the residuals' mean in FRAP model.
α | 0.01 | 0.05 | 0.1 | 0.2 |
---|---|---|---|---|
t-value | 2.8120 | 2.0175 | 1.6820 | 1.3020 |
Decision | Accept H _{0} | Accept H _{0} | Accept H _{0} | Accept H _{0} |
The rejection region is those values of T > ${\chi}_{(\alpha ,k-1)}^{2}$ in which ${\chi}_{(\alpha ,k-1)}^{2}$ is the upper critical value of the chi-square distribution with k - 1 degree of freedom at the level of significance α.
The pooled weighted variance was found to be ${S}_{p}^{2}$ = 7.1030 × 10^{-4}. The Bartlett's statistic was calculated as T = 9.5454 which is less than the upper critical value of the χ^{2} for two degrees of freedom (k = 3) at one per cent level of significance (${\chi}_{(0.01,2)}^{2}$ = 10.60). It is, however, more than the tabled value for five per cent level of significance. At one percent level of significance the null hypothesis (the residuals have constant variance) cannot be rejected. Based on this test and analysis of residual plot versus laser beam recovery, it is concluded that the residuals have equal variance.
in which r _{ s }is the correlation coefficient in the sample.
The null hypothesis (correlation coefficient is zero) is rejected when the absolute value of the t-statistic is greater than the critical t-value (t < -t _{ α/2}or t _{ α/2}<t) at the level of significance α.
The residuals were first divided into two sub-groups: r _{1} = r(1 : end - 1) r _{2} = r(2 : end)
The results of hypothesis test on the correlation of residuals in FRAP model.
α | 0.01 | 0.05 | 0.1 | 0.2 |
---|---|---|---|---|
t-value | 2.7040 | 2.0210 | 1.6820 | 1.3030 |
Decision | Accept H _{0} | Accept H _{0} | Accept H _{0} | Accept H _{0} |
where k is the number of intervals or cells.
or; H _{0} : r = N(0.0012,0.0224) H _{ A }: r ≠ N(0.0012,0.0224)
where μ and σ are the mean and the standard deviation of the residuals in population.
Since the calculated χ^{2} (3.6190) is less than the tabled value (${\chi}_{(0.80,2)}^{2}$ = 4.61), the null hypothesis (the residuals are normally distributed) cannot be rejected even at 20 per cent level of significance implying that the residuals are normally distributed.
The chi-square test is a powerful test when the sample size is large. However, combining cells when the expected error frequencies are less than five loses information and hence decreases the power of the test. Furthermore, for very small samples this test is not applicable [49]. To overcome these limitations, the Kolmogorov-Smirnov one sample test is usually used since it treats each observation separately and does not lose information through merging of categories. This test is more powerful than the chi-square test when sample size is not large.
The Kolmogorov-Smirnov one sample test was used to verify if the residuals are normally distributed. Results (not shown here) indicate that the null hypothesis (the errors are normally distributed) cannot be rejected even at 20 per cent level of significance implying that the residuals are strongly normally distributed.
In conclusion, detailed residual analysis indicates that: 1) residuals have zero mean, 2) residuals have constant variance, 3) residuals are normally distributed, and 4) residuals are uncorrelated. Therefore, the necessary and sufficient criteria for least square parameter optimization, which was used in this study, were met.
Conclusion
A Galerkin-based finite element model was developed and applied to solve a system of two coupled partial differential equations governing GFP-GR transport and reaction in living cells. A finite domain was used to formulate boundary conditions on the cell membrane. The simulator was coupled with the Levenberg-Marquardt algorithm and a FRAP time series to estimate protein mass transport and reaction rate parameters. The developed strategy presents excellent agreement with the experimental data and the developed strategy is an efficient approach to extract as much physiochemical information from the FRAP protocol as possible. Uniqueness analysis of the inverse problem indicates that the FRAP protocol provides insufficient information for unique quantification of diffusion coefficient and binding rate parameters.
Declarations
Acknowledgements
This study was supported in part by the U.S. Department of Energy under Grant No. DOEG0204ER25619. The authors would like to thank Dr. J. McNally head of the Receptor Biology and Gene Expression Laboratory at the National Cancer Institute, Bethesda, MD for providing the experimental data.
Authors’ Affiliations
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